---
layout: post
title: "Data manipulation with numpy: tips and tricks, part 1"
date: "2015-09-29"
author: Alex Rogozhnikov
tags:
- notebook
---


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<h1 id="Data-manipulation-with-numpy:-tips-and-tricks,-part-1">Data manipulation with numpy: tips and tricks, part 1<a class="anchor-link" href="#Data-manipulation-with-numpy:-tips-and-tricks,-part-1">&#182;</a></h1><p>Some inobvious examples of what you can do with numpy are collected here.</p>
<p>Examples are mostly coming from area of machine learning, but will be useful if you're doing number crunching in python.</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">print_function</span> <span class="c"># for python 2 &amp; python 3 compatibility</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
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<h2 id="1.-numpy.argsort">1. numpy.argsort<a class="anchor-link" href="#1.-numpy.argsort">&#182;</a></h2>
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<h3 id="Sorting-values-of-one-array-according-to-the-other">Sorting values of one array according to the other<a class="anchor-link" href="#Sorting-values-of-one-array-according-to-the-other">&#182;</a></h3><p>Say, we want to order the people according to their age and their heights.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">ages</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">60</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<span class="n">heights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">210</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>

<span class="k">print</span><span class="p">(</span><span class="n">ages</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">heights</span><span class="p">)</span>
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<pre>[49 45 44 52 44 57 46 49 31 50]
[209 183 202 188 205 179 209 187 156 209]
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<div class=" highlight hl-ipython2"><pre><span class="n">sorter</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">ages</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">ages</span><span class="p">[</span><span class="n">sorter</span><span class="p">])</span>
<span class="k">print</span><span class="p">(</span><span class="n">heights</span><span class="p">[</span><span class="n">sorter</span><span class="p">])</span>
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<pre>[31 44 44 45 46 49 49 50 52 57]
[156 202 205 183 209 209 187 209 188 179]
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<p>once you computed permutation, you can apply it many times - this is fast.</p>
<p>Frequently to solve this problem people use sorted(zip(ages, heights)), which is much slower (10-20 times slower on large arrays).</p>

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<h3 id="Computing-inverse-of-permutation">Computing inverse of permutation<a class="anchor-link" href="#Computing-inverse-of-permutation">&#182;</a></h3><p>permutations in numpy are simply arrays:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">permutation</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">permutation</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">original</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="s">&#39;abcdefghij&#39;</span><span class="p">))</span>

<span class="k">print</span><span class="p">(</span><span class="n">permutation</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">original</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">original</span><span class="p">[</span><span class="n">permutation</span><span class="p">])</span>
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<pre>[1 7 4 9 2 8 0 5 6 3]
[&apos;a&apos; &apos;b&apos; &apos;c&apos; &apos;d&apos; &apos;e&apos; &apos;f&apos; &apos;g&apos; &apos;h&apos; &apos;i&apos; &apos;j&apos;]
[&apos;b&apos; &apos;h&apos; &apos;e&apos; &apos;j&apos; &apos;c&apos; &apos;i&apos; &apos;a&apos; &apos;f&apos; &apos;g&apos; &apos;d&apos;]
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<p><strong>Inverse permutation</strong> is computed using numpy.argsort (again!)</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">inverse_permutation</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">permutation</span><span class="p">)</span>

<span class="k">print</span><span class="p">(</span><span class="n">original</span><span class="p">[</span><span class="n">permutation</span><span class="p">][</span><span class="n">inverse_permutation</span><span class="p">])</span>
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<pre>[&apos;a&apos; &apos;b&apos; &apos;c&apos; &apos;d&apos; &apos;e&apos; &apos;f&apos; &apos;g&apos; &apos;h&apos; &apos;i&apos; &apos;j&apos;]
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<p><strong>This is true because of two facts</strong>:</p>
<ol>
<li><p>indexing operation is associative (dramatically simple and interesting fact):</p>

<pre><code>a[b][c] = a[b[c]]</code></pre>
<p>provided that <code>a</code>, <code>b</code>, <code>c</code> are 1-dimensional arrays</p>
</li>
<li><p>`permutation[inverse_permutation] is identical permutation:</p>
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<div class=" highlight hl-ipython2"><pre><span class="n">permutation</span><span class="p">[</span><span class="n">inverse_permutation</span><span class="p">]</span>
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<pre>array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])</pre>
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<h3 id="Even-faster-inverse-permutation">Even faster inverse permutation<a class="anchor-link" href="#Even-faster-inverse-permutation">&#182;</a></h3><p>As was said, numpy.argsort returns inverse permutation, but it takes $O(n \log(n))$ time, while computing inverse permutation should take $O(n)$.</p>
<p>This optimal way can be written in numpy.</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">permutation</span><span class="p">))</span>

<span class="n">inverse_permutation</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">permutation</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">int</span><span class="p">)</span>
<span class="n">inverse_permutation</span><span class="p">[</span><span class="n">permutation</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">permutation</span><span class="p">))</span>
<span class="k">print</span><span class="p">(</span><span class="n">inverse_permutation</span><span class="p">)</span>
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<pre>[6 0 4 9 2 7 8 1 5 3]
[6 0 4 9 2 7 8 1 5 3]
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<h3 id="Computing-order-of-elements-in-array">Computing order of elements in array<a class="anchor-link" href="#Computing-order-of-elements-in-array">&#182;</a></h3><p>frequently it is important to compute order of each value in array.</p>
<p>In other words, for each element in array we want to find the number of elements smaller than given.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>

<span class="k">print</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">data</span><span class="p">)))</span>
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<pre>[ 0.69378073  0.47532658  0.11610735  0.89700047  0.04700985  0.24701576
  0.29017543  0.27857828  0.62900998  0.08187479]
[8 6 2 9 0 3 5 4 7 1]
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<p><strong>NB:</strong> there is scipy function which does the same, but it's more general and faster, so prefer using it:</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">rankdata</span>
<span class="n">rankdata</span><span class="p">(</span><span class="n">data</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
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<pre>array([ 8.,  6.,  2.,  9.,  0.,  3.,  5.,  4.,  7.,  1.])</pre>
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<h3 id="IronTransform-(flattener-of-distribution)">IronTransform (flattener of distribution)<a class="anchor-link" href="#IronTransform-(flattener-of-distribution)">&#182;</a></h3><p>Sometimes you need to write monotonic tranformation, which turns one distribution into uniform.</p>
<p>This method is useful to compare distributions or to work with distributions with heavy tails or strange shape.</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">class</span> <span class="nc">IronTransform</span><span class="p">:</span>
    <span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">):</span>
        <span class="n">weights</span> <span class="o">=</span> <span class="n">weights</span> <span class="o">/</span> <span class="n">weights</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
        <span class="n">sorter</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">sorter</span><span class="p">]</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">weights</span><span class="p">[</span><span class="n">sorter</span><span class="p">])</span>
        <span class="k">return</span> <span class="bp">self</span>
        
    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="p">):</span>
        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
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<p>Let's apply <strong>Iron</strong> transformation to data</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">sig_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">bck_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">sig_pred</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">bck_pred</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Now we can see that <strong>IronTransform</strong> actually was able to turn signal distribution to uniform:</p>

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<div class="prompt input_prompt">In&nbsp;[13]:</div>
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<div class=" highlight hl-ipython2"><pre><span class="n">iron</span> <span class="o">=</span> <span class="n">IronTransform</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">sig_pred</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sig_pred</span><span class="p">)))</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">iron</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">sig_pred</span><span class="p">),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">iron</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">bck_pred</span><span class="p">),</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<h3 id="How-IronTransform-works">How IronTransform works<a class="anchor-link" href="#How-IronTransform-works">&#182;</a></h3><p>To turn any distribution into uniform, you should use mapping:
$x \to F(x)$, here $F(x)$ is cumulative distribution function.</p>
<p>In other words, for each point $x$ we should compute the part of distribution to the left. 
This was done by summing all weights to this point using <code>numpy.cumsum</code> (we also had to normalize weights).</p>
<p>To use the learned mapping, linear interpolation (<code>numpy.interp</code>) was applied.
We can visualize this mapping:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">iron</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">iron</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
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<pre>[&lt;matplotlib.lines.Line2D at 0x108df0750&gt;]</pre>
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<h2 id="IronTransform-and-histogram-equalization">IronTransform and histogram equalization<a class="anchor-link" href="#IronTransform-and-histogram-equalization">&#182;</a></h2><p>The same method is applied in computer vision to 'improve' low-contrast images and called <a href="https://en.wikipedia.org/wiki/Histogram_equalization">histogram equalization</a>.</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">PIL</span> <span class="kn">import</span> <span class="n">Image</span>
<span class="n">image</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">Image</span><span class="o">.</span><span class="n">open</span><span class="p">(</span><span class="s">&#39;AquaTermi_lowcontrast.jpg&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">))</span>
<span class="n">flat_image</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">IronTransform</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">flat_image</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">flat_image</span><span class="p">)))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">image</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">14</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">),</span> <span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s">&#39;gray&#39;</span><span class="p">),</span> <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">&#39;original&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">),</span> <span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s">&#39;gray&#39;</span><span class="p">),</span> <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">&#39;after histogram equalization&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<p>Also worth mentioning: to fight very small / large numbers, use <code>numpy.clip</code>. Simple and very handy:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">clip</span><span class="p">([</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="n">a_min</span><span class="o">=-</span><span class="mi">15</span><span class="p">,</span> <span class="n">a_max</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
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<pre>array([-15, -10,   0,  10,  15])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
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<pre>[-5 -4 -3 -2 -1  0  1  2  3  4]
[0 0 0 0 0 0 1 2 3 4]
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<h1 id="2.-Broadcasting,-numpy.newaxis">2. Broadcasting, numpy.newaxis<a class="anchor-link" href="#2.-Broadcasting,-numpy.newaxis">&#182;</a></h1><p><a href="http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html">Broadcasting</a> is very useful trick, which simplifies many operations.</p>

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<h3 id="Weighted-covariation-matrix">Weighted covariation matrix<a class="anchor-link" href="#Weighted-covariation-matrix">&#182;</a></h3><p>numpy has <code>cov</code> function, but it doesn't support weights of samples. Let's write our own covariation matrix.</p>
<p>Let $X_{ij}$ be the original data. $i$ is indexing samples, $j$ is indexing features.
$COV_{jk} = \sum_i X_{ij} w_i X_{ik}$</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">covariation</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">):</span>
    <span class="n">weights</span> <span class="o">=</span> <span class="n">weights</span> <span class="o">/</span> <span class="n">weights</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
    <span class="k">return</span> <span class="n">data</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">weights</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span> <span class="o">*</span> <span class="n">data</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
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<pre>array([[ 0.98246422, -0.09213933,  0.04220243, -0.05215392,  0.07939284],
       [-0.09213933,  1.22118146, -0.10138206,  0.23070906,  0.14642467],
       [ 0.04220243, -0.10138206,  0.84470403, -0.00338101, -0.1249949 ],
       [-0.05215392,  0.23070906, -0.00338101,  1.10425395, -0.11689576],
       [ 0.07939284,  0.14642467, -0.1249949 , -0.11689576,  1.37570908]])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="n">covariation</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)))</span>
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<pre>array([[  9.80919618e-01,  -9.15155980e-02,   3.81764158e-02,
         -5.88816111e-02,   8.46305808e-02],
       [ -9.15155980e-02,   1.20898034e+00,  -1.00238681e-01,
          2.28662578e-01,   1.44743583e-01],
       [  3.81764158e-02,  -1.00238681e-01,   8.37825668e-01,
         -1.91880751e-04,  -1.26370312e-01],
       [ -5.88816111e-02,   2.28662578e-01,  -1.91880751e-04,
          1.09955816e+00,  -1.21007564e-01],
       [  8.46305808e-02,   1.44743583e-01,  -1.26370312e-01,
         -1.21007564e-01,   1.36634581e+00]])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="n">covariation</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">)</span>
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<pre>array([[ 1.08429417, -0.11032936,  0.09253106, -0.006654  ,  0.14981251],
       [-0.11032936,  1.04538348, -0.02432767,  0.22118083, -0.06601326],
       [ 0.09253106, -0.02432767,  0.95356108, -0.05339915, -0.08393104],
       [-0.006654  ,  0.22118083, -0.05339915,  1.09609127, -0.17836458],
       [ 0.14981251, -0.06601326, -0.08393104, -0.17836458,  1.22027334]])</pre>
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<p>alternative way to do this is via <code>numpy.einsum</code>:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s">&#39;ij, ik, i -&gt; jk&#39;</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">weights</span> <span class="o">/</span> <span class="n">weights</span><span class="o">.</span><span class="n">sum</span><span class="p">())</span>
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<pre>array([[ 1.08429417, -0.11032936,  0.09253106, -0.006654  ,  0.14981251],
       [-0.11032936,  1.04538348, -0.02432767,  0.22118083, -0.06601326],
       [ 0.09253106, -0.02432767,  0.95356108, -0.05339915, -0.08393104],
       [-0.006654  ,  0.22118083, -0.05339915,  1.09609127, -0.17836458],
       [ 0.14981251, -0.06601326, -0.08393104, -0.17836458,  1.22027334]])</pre>
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<h3 id="Pearson-correlation">Pearson correlation<a class="anchor-link" href="#Pearson-correlation">&#182;</a></h3><p>One more example of broadcasting: computation of Pearson coefficient.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
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<pre>array([[ 1.        , -0.08411948,  0.04632621, -0.0500719 ,  0.0682904 ],
       [-0.08411948,  1.        , -0.09982027,  0.19867356,  0.1129694 ],
       [ 0.04632621, -0.09982027,  1.        , -0.00350074, -0.11595159],
       [-0.0500719 ,  0.19867356, -0.00350074,  1.        , -0.09484206],
       [ 0.0682904 ,  0.1129694 , -0.11595159, -0.09484206,  1.        ]])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">my_corrcoef</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">):</span>
    <span class="n">data</span> <span class="o">=</span> <span class="n">data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">weights</span><span class="p">)</span>
    <span class="n">shifted_cov</span> <span class="o">=</span> <span class="n">covariation</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">weights</span><span class="p">)</span>
    <span class="n">cov_diag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">shifted_cov</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">shifted_cov</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">cov_diag</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span> <span class="o">*</span> <span class="n">cov_diag</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:])</span>

<span class="n">my_corrcoef</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)))</span>
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<pre>array([[ 1.        , -0.08411948,  0.04632621, -0.0500719 ,  0.0682904 ],
       [-0.08411948,  1.        , -0.09982027,  0.19867356,  0.1129694 ],
       [ 0.04632621, -0.09982027,  1.        , -0.00350074, -0.11595159],
       [-0.0500719 ,  0.19867356, -0.00350074,  1.        , -0.09484206],
       [ 0.0682904 ,  0.1129694 , -0.11595159, -0.09484206,  1.        ]])</pre>
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<h3 id="Pairwise-distances-between-vectors">Pairwise distances between vectors<a class="anchor-link" href="#Pairwise-distances-between-vectors">&#182;</a></h3><p>we're given a set of vectors (say, 1000 vectors). The problem is to compute their pairwise distances.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">[</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="n">distances</span> <span class="o">=</span> <span class="p">((</span><span class="n">X</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span> <span class="o">-</span> <span class="n">X</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:])</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="mf">0.5</span>
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<p>also, since algebra is you friend, you can do it times faster: 
$$||x_i - x_j||^2 = ||x_i||^2 + ||x_j||^2 - 2 (x_i, x_j) $$
so dot products is the only thing you need.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">products</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="n">distances2</span> <span class="o">=</span> <span class="n">products</span><span class="o">.</span><span class="n">diagonal</span><span class="p">()[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span> <span class="o">+</span> <span class="n">products</span><span class="o">.</span><span class="n">diagonal</span><span class="p">()[</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">,</span> <span class="p">:]</span>  <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">products</span>
<span class="n">distances2</span> <span class="o">**=</span> <span class="mf">0.5</span>
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<p>... but keep in mind there is <strong>sklearn.metrics.pairwise</strong> which does it for you and has different options</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">sklearn.metrics.pairwise</span> <span class="kn">import</span> <span class="n">pairwise_distances</span>
<span class="n">distances_sklearn</span> <span class="o">=</span> <span class="n">pairwise_distances</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

<span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">distances</span><span class="p">,</span> <span class="n">distances_sklearn</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">distances2</span><span class="p">,</span> <span class="n">distances_sklearn</span><span class="p">)</span>
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<pre>(True, True)</pre>
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<h1 id="3.-numpy.unique,-numpy.searchsorted">3. numpy.unique, numpy.searchsorted<a class="anchor-link" href="#3.-numpy.unique,-numpy.searchsorted">&#182;</a></h1>
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<h3 id="Converting-categories-to-small-integers">Converting categories to small integers<a class="anchor-link" href="#Converting-categories-to-small-integers">&#182;</a></h3><p>Despite categories are easier to read, operating with them is much faster after you represent them as small numbers <br /> (for instance, in range [0, n_categories - 1])</p>
<p>First generate some categories:</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</span>
<span class="n">possible_categories</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="n">combinations</span><span class="p">(</span><span class="s">&#39;abcdefghijklmn&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
<span class="n">categories</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">possible_categories</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">categories</span><span class="p">)</span>
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<pre>[&apos;cd&apos; &apos;bi&apos; &apos;ag&apos; ..., &apos;gn&apos; &apos;ik&apos; &apos;kl&apos;]
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<p>Now we use <strong>numpy.unique</strong> to convert them to numbers (pay attention to <code>return_inverse=True</code>)</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">unique_categories</span><span class="p">,</span> <span class="n">new_categories</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">categories</span><span class="p">,</span> <span class="n">return_inverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="k">print</span><span class="p">(</span><span class="n">unique_categories</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">new_categories</span><span class="p">)</span>
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<pre>[&apos;ab&apos; &apos;ac&apos; &apos;ad&apos; &apos;ae&apos; &apos;af&apos; &apos;ag&apos; &apos;ah&apos; &apos;ai&apos; &apos;aj&apos; &apos;ak&apos; &apos;al&apos; &apos;am&apos; &apos;an&apos; &apos;bc&apos; &apos;bd&apos;
 &apos;be&apos; &apos;bf&apos; &apos;bg&apos; &apos;bh&apos; &apos;bi&apos; &apos;bj&apos; &apos;bk&apos; &apos;bl&apos; &apos;bm&apos; &apos;bn&apos; &apos;cd&apos; &apos;ce&apos; &apos;cf&apos; &apos;cg&apos; &apos;ch&apos;
 &apos;ci&apos; &apos;cj&apos; &apos;ck&apos; &apos;cl&apos; &apos;cm&apos; &apos;cn&apos; &apos;de&apos; &apos;df&apos; &apos;dg&apos; &apos;dh&apos; &apos;di&apos; &apos;dj&apos; &apos;dk&apos; &apos;dl&apos; &apos;dm&apos;
 &apos;dn&apos; &apos;ef&apos; &apos;eg&apos; &apos;eh&apos; &apos;ei&apos; &apos;ej&apos; &apos;ek&apos; &apos;el&apos; &apos;em&apos; &apos;en&apos; &apos;fg&apos; &apos;fh&apos; &apos;fi&apos; &apos;fj&apos; &apos;fk&apos;
 &apos;fl&apos; &apos;fm&apos; &apos;fn&apos; &apos;gh&apos; &apos;gi&apos; &apos;gj&apos; &apos;gk&apos; &apos;gl&apos; &apos;gm&apos; &apos;gn&apos; &apos;hi&apos; &apos;hj&apos; &apos;hk&apos; &apos;hl&apos; &apos;hm&apos;
 &apos;hn&apos; &apos;ij&apos; &apos;ik&apos; &apos;il&apos; &apos;im&apos; &apos;in&apos; &apos;jk&apos; &apos;jl&apos; &apos;jm&apos; &apos;jn&apos; &apos;kl&apos; &apos;km&apos; &apos;kn&apos; &apos;lm&apos; &apos;ln&apos;
 &apos;mn&apos;]
[25 19  5 ..., 69 77 85]
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<p>Also we can reconstruct old categories if needed:</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">print</span><span class="p">(</span><span class="n">categories</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">unique_categories</span><span class="p">[</span><span class="n">new_categories</span><span class="p">])</span>
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<pre>[&apos;cd&apos; &apos;bi&apos; &apos;ag&apos; ..., &apos;gn&apos; &apos;ik&apos; &apos;kl&apos;]
[&apos;cd&apos; &apos;bi&apos; &apos;ag&apos; ..., &apos;gn&apos; &apos;ik&apos; &apos;kl&apos;]
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<h3 id="Mapping-new-categories">Mapping new categories<a class="anchor-link" href="#Mapping-new-categories">&#182;</a></h3><p>When we need to map new categories using the same lookup table, we can use <code>numpy.searchsorted</code></p>

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<div class=" highlight hl-ipython2"><pre><span class="n">categories2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">possible_categories</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
<span class="n">new_categories2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">unique_categories</span><span class="p">,</span> <span class="n">categories2</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">categories2</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">unique_categories</span><span class="p">[</span><span class="n">new_categories2</span><span class="p">])</span>
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<pre>[&apos;gn&apos; &apos;hn&apos; &apos;be&apos; ..., &apos;il&apos; &apos;bk&apos; &apos;bl&apos;]
[&apos;gn&apos; &apos;hn&apos; &apos;be&apos; ..., &apos;il&apos; &apos;bk&apos; &apos;bl&apos;]
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<h3 id="Checking-values-present-in-the-lookup">Checking values present in the lookup<a class="anchor-link" href="#Checking-values-present-in-the-lookup">&#182;</a></h3><p>Usually, we use <code>set</code> to have many checks that each element exists in array. But <strong>numpy.unique</strong> is better option - it works dozens times faster.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">in1d</span><span class="p">([</span><span class="s">&#39;ab&#39;</span><span class="p">,</span> <span class="s">&#39;ac&#39;</span><span class="p">,</span> <span class="s">&#39;something new&#39;</span><span class="p">],</span> <span class="n">unique_categories</span><span class="p">)</span>
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<pre>array([ True,  True, False], dtype=bool)</pre>
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<h3 id="Handling-missing-categories">Handling missing categories<a class="anchor-link" href="#Handling-missing-categories">&#182;</a></h3><p>When some new category appears in test data, this is always a trouble. Possible solutions:</p>
<ul>
<li>map randomly to one of present categories (this, i.e. is done by hashing trick)</li>
<li>make a special category 'unknown' where all new values will be mapped. Then special value may be chosen for it.
This approach is more general, but requires more efforts.</li>
</ul>
<p>Below you can find the transformer for second approach:</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">class</span> <span class="nc">CategoryMapper</span><span class="p">:</span>
    <span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">categories</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">lookup</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">categories</span><span class="p">)</span>
        <span class="k">return</span> <span class="bp">self</span>
        
    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">categories</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Converts categories to numbers, 0 is reserved for new values (not present in fitted data)&quot;&quot;&quot;</span>
        <span class="k">return</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lookup</span><span class="p">,</span> <span class="n">categories</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">in1d</span><span class="p">(</span><span class="n">categories</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">lookup</span><span class="p">)</span>    
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<div class=" highlight hl-ipython2"><pre><span class="n">CategoryMapper</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">categories</span><span class="p">)</span><span class="o">.</span><span class="n">transform</span><span class="p">([</span><span class="n">unique_categories</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s">&#39;abc&#39;</span><span class="p">])</span>
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<pre>array([1, 0])</pre>
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<h1 id="4.-numpy.percentile,-numpy.bincount">4. numpy.percentile, numpy.bincount<a class="anchor-link" href="#4.-numpy.percentile,-numpy.bincount">&#182;</a></h1>
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<h3 id="Splitting-data-into-bins">Splitting data into bins<a class="anchor-link" href="#Splitting-data-into-bins">&#182;</a></h3><p>Splitting data into bins is frequently necessary. Simplest way done e.g. by <code>numpy.histogram</code> is splitting region with even bins of same size.</p>
<p>This usually works poorly with distributions with long tails (or strange shape) and requires additional transformation to avoid empty or too large bins.</p>
<p>However, for most cases creating the bins with equal number of events gives more convenient and stable results.
Selecting the edges of bins can be done with <code>numpy.percentile</code>.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">20000</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">40</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">edges</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="n">edges</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
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<p>the number of events within each bin is equal now.</p>

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<h3 id="Defining-the-index-of-bin">Defining the index of bin<a class="anchor-link" href="#Defining-the-index-of-bin">&#182;</a></h3><p>Use <code>numpy.searchsorted</code> to compute the index of bin for each value in <code>x</code>. (<code>numpy.digitize</code> is the other option, it does the same).</p>
<p><code>numpy.searchsorted</code> assumes that first array is sorted and uses binary search, so it is effective even for large amount of bins.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">searchsorted</span><span class="p">(</span><span class="n">edges</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
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<pre>array([ 5,  2, 16, ...,  7,  9,  3])</pre>
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<h3 id="Counter">Counter<a class="anchor-link" href="#Counter">&#182;</a></h3><p>Need to know number of occurences for each category? Use <code>numpy.bincount</code> to compute the statistics:</p>

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<pre>array([25, 19,  5, ..., 69, 77, 85])</pre>
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<p>This will compute how many times each category present in the data:</p>

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<pre>array([112, 105, 127, 112, 118, 107, 129,  94, 118, 105, 109, 103,  97,
       108, 110,  97, 103, 111, 103, 102, 108, 115, 121,  98, 121, 115,
       105, 113, 109,  97,  91, 109, 109, 107, 114, 123,  99,  98,  92,
       117, 103,  94, 111, 113, 125,  98, 110, 122, 108, 117, 106, 100,
       111, 108, 106, 130, 116,  97, 112, 106, 103, 103, 109, 107, 106,
       121, 107, 117, 115, 113, 109, 107, 122, 123, 117, 104,  99, 125,
       110, 122,  93, 105, 112, 118, 116, 105, 110, 112, 123, 116, 127])</pre>
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<p>the same done with <code>collections.Counter</code> (which is much slower)</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>
<span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">Counter</span><span class="p">(</span><span class="n">new_categories</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">()))[:,</span> <span class="mi">1</span><span class="p">]</span>
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<pre>array([112, 105, 127, 112, 118, 107, 129,  94, 118, 105, 109, 103,  97,
       108, 110,  97, 103, 111, 103, 102, 108, 115, 121,  98, 121, 115,
       105, 113, 109,  97,  91, 109, 109, 107, 114, 123,  99,  98,  92,
       117, 103,  94, 111, 113, 125,  98, 110, 122, 108, 117, 106, 100,
       111, 108, 106, 130, 116,  97, 112, 106, 103, 103, 109, 107, 106,
       121, 107, 117, 115, 113, 109, 107, 122, 123, 117, 104,  99, 125,
       110, 122,  93, 105, 112, 118, 116, 105, 110, 112, 123, 116, 127])</pre>
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<h3 id="Replace-category-with-number-of-occurences">Replace category with number of occurences<a class="anchor-link" href="#Replace-category-with-number-of-occurences">&#182;</a></h3><p>There are several tricks important to process the categorical features. 
Most known one is one-hot encoding, but let's speak about others.</p>
<p>One of tricks is to replace in each event category (which cannot be processed by most algorithms) with a number of times it presents in data.</p>
<p>This can be written in <strong>numpy</strong> one-liner:</p>

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<pre>array([115, 102, 107, ..., 113, 125, 105])</pre>
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<h3 id="Averaging-predictions-over-category">Averaging predictions over category<a class="anchor-link" href="#Averaging-predictions-over-category">&#182;</a></h3><p>Another important trick: replace category with average prediction for events from this category.</p>
<p>This is more tricky, since we need to add weights to <code>numpy.bincount</code>:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">means_over_category</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">new_categories</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">predictions</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">new_categories</span><span class="p">)</span>
<span class="n">means_over_category</span><span class="p">[</span><span class="n">new_categories</span><span class="p">]</span>
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<pre>array([ 0.03873642, -0.00404972,  0.01078471, ..., -0.13723158,
       -0.13214773,  0.12129524])</pre>
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<h3 id="Variation-of-predictions-over-category">Variation of predictions over category<a class="anchor-link" href="#Variation-of-predictions-over-category">&#182;</a></h3><p>The same trick can be applied to measure deviation after invloving some math:
$$ \mathbb{V}x = \mathbb{E}x^2 - (\mathbb{E}x)^2 $$</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">means_of_squares_over_category</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">new_categories</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">predictions</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">new_categories</span><span class="p">)</span>
<span class="n">var_over_category</span> <span class="o">=</span> <span class="n">means_of_squares_over_category</span> <span class="o">-</span> <span class="n">means_over_category</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">var_over_category</span><span class="p">[</span><span class="n">new_categories</span><span class="p">]</span>
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<pre>array([ 0.85843132,  0.94388082,  0.96350536, ...,  0.8831292 ,
        1.01031933,  0.67702705])</pre>
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<p>You can also assign weights to events and compute weighted mean / weighted variation using the same functions.</p>

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<h2 id="5.-Numpy.ufunc.at">5. Numpy.ufunc.at<a class="anchor-link" href="#5.-Numpy.ufunc.at">&#182;</a></h2><p>To proceed futher, we need to know that there is class of binary (and unary) functions: <code>numpy.ufuncs</code></p>
<p>Examples: <code>numpy.add</code>, <code>numpy.subtract</code>, <code>numpy.multiply</code>, <code>numpy.maximum</code>, <code>numpy.minimum</code></p>
<p>In the following example we cook a new column with maximal predictions over category using <code>ufunc.at</code></p>

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<div class=" highlight hl-ipython2"><pre><span class="n">max_over_category</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">new_categories</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">infty</span>
<span class="n">np</span><span class="o">.</span><span class="n">maximum</span><span class="o">.</span><span class="n">at</span><span class="p">(</span><span class="n">max_over_category</span><span class="p">,</span> <span class="n">new_categories</span><span class="p">,</span> <span class="n">predictions</span><span class="p">)</span>

<span class="k">print</span><span class="p">(</span><span class="n">max_over_category</span><span class="p">[</span><span class="n">new_categories</span><span class="p">])</span>
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<pre>[ 2.87279069  2.43059696  2.04921886 ...,  2.62513163  2.81520326
  1.72020708]
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<p>Important note: <code>numpy.add.at</code> and <code>numpy.bincount</code> are very similar, but latter is dozens of times faster. Always prefer it to <code>numpy.add.at</code></p>

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<div class=" highlight hl-ipython2"><pre><span class="o">%</span><span class="k">timeit</span> np.add.at(max_over_category, new_categories, predictions)
<span class="o">%</span><span class="k">timeit</span> np.bincount(new_categories, weights=predictions)
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<pre>100 loops, best of 3: 1.63 ms per loop
10000 loops, best of 3: 33 µs per loop
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<h3 id="Multidimensional-statistics:-numpy.add.at-and-smart-indexing">Multidimensional statistics: numpy.add.at and smart indexing<a class="anchor-link" href="#Multidimensional-statistics:-numpy.add.at-and-smart-indexing">&#182;</a></h3><p>When need to compute number of times each combination met, there are two ways:</p>
<ol>
<li>convert couple of variables to new variable, e.g. by feature1 * (numpy.max(feature2) + 1) + feature2</li>
<li>create multidimensional table</li>
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<p>I prefer first, code below is demonstration of second approach and it's benefits.</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">first_category</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_categories</span><span class="p">))</span>
<span class="n">second_category</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_categories</span><span class="p">))</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">counters</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">first_category</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">second_category</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">np</span><span class="o">.</span><span class="n">add</span><span class="o">.</span><span class="n">at</span><span class="p">(</span><span class="n">counters</span><span class="p">,</span> <span class="p">[</span><span class="n">first_category</span><span class="p">,</span> <span class="n">second_category</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
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<p>Now some useful things we can do with colected statistic. First thing is create feature, for which we use fancy indexing:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">counters</span><span class="p">[</span><span class="n">first_category</span><span class="p">,</span> <span class="n">second_category</span><span class="p">]</span>
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<pre>array([ 1.,  1.,  3., ...,  4.,  3.,  1.])</pre>
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<p>We can also infer different statistics like:</p>

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<div class=" highlight hl-ipython2"><pre><span class="c"># occurences of second category</span>
<span class="n">counters</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)[</span><span class="n">second_category</span><span class="p">]</span>
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<pre>array([ 122.,  102.,   82., ...,  112.,  118.,  100.])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="c"># maximal occurences of second category with same value of first category</span>
<span class="n">counters</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)[</span><span class="n">second_category</span><span class="p">]</span>
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<pre>array([ 4.,  4.,  3., ...,  4.,  4.,  7.])</pre>
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<div class=" highlight hl-ipython2"><pre><span class="c"># number of occurences of second category with fixed first category:</span>
<span class="n">counters</span><span class="p">[</span><span class="mi">42</span><span class="p">,</span> <span class="n">second_category</span><span class="p">]</span>
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<pre>array([ 0.,  0.,  0., ...,  4.,  4.,  1.])</pre>
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<ul>
<li>you are not limited to <code>numpy.add</code>, remember there are other ufuncs.</li>
<li>Few space to keep whole matrix? Use <code>scipy.sparse</code> if there are too many zero elements. 
Alternatively, you can use matrix decompositions to keep approximation of matrix and compute result on-the-fly.</li>
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<h3 id="Computing-ROC-curve">Computing ROC curve<a class="anchor-link" href="#Computing-ROC-curve">&#182;</a></h3><p>As an important exercise, let's implement the computation of ROC curve. (interactive demo of ROC curve may be found <a href="https://arogozhnikov.github.io/2015/10/05/roc-curve.html">here</a> )</p>
<p>We will write the same as <code>sklearn.metrics.roc_curve</code> (actually, simplified version of it).</p>
<p>ROC curve takes three arrays: labels (binary, 0 or 1), predictions (real-valued) and weights.</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">roc_curve</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">predictions</span><span class="p">,</span> <span class="n">sample_weight</span><span class="p">):</span>
    <span class="n">sig_weights</span> <span class="o">=</span> <span class="n">sample_weight</span> <span class="o">*</span> <span class="p">(</span><span class="n">labels</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">bck_weights</span> <span class="o">=</span> <span class="n">sample_weight</span> <span class="o">*</span> <span class="p">(</span><span class="n">labels</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
    <span class="n">thresholds</span><span class="p">,</span> <span class="n">predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">predictions</span><span class="p">,</span> <span class="n">return_inverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
    <span class="n">tpr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">predictions</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">sig_weights</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cumsum</span><span class="p">()</span>
    <span class="n">fpr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">predictions</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="n">bck_weights</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cumsum</span><span class="p">()</span>
    <span class="n">tpr</span> <span class="o">/=</span> <span class="n">tpr</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
    <span class="n">fpr</span> <span class="o">/=</span> <span class="n">fpr</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
    <span class="k">return</span> <span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="n">thresholds</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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<span class="n">size</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">size</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="mi">1</span>
<span class="n">predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span> <span class="o">+</span> <span class="n">labels</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
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<p>Let's check how result looks like:</p>

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<div class=" highlight hl-ipython2"><pre><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="n">labels</span> <span class="o">==</span> <span class="mi">0</span><span class="p">],</span> <span class="n">bins</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="n">labels</span> <span class="o">==</span> <span class="mi">1</span><span class="p">],</span> <span class="n">bins</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">fpr1</span><span class="p">,</span> <span class="n">tpr1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">&#39;FPR&#39;</span><span class="p">),</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&#39;TPR&#39;</span><span class="p">)</span>
</pre></div>

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<div class="output_area"><div class="prompt output_prompt">Out[59]:</div>


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<pre>(&lt;matplotlib.text.Text at 0x109956e90&gt;, &lt;matplotlib.text.Text at 0x10995c190&gt;)</pre>
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<div class=" highlight hl-ipython2"><pre><span class="c"># sometimes sklearn adds one more element to all arrays.</span>
<span class="c"># I have no idea why this is needed and when, but in this case all answers turn to False</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">roc_curve</span> <span class="k">as</span> <span class="n">sklearn_roc_curve</span>
<span class="n">fpr2</span><span class="p">,</span> <span class="n">tpr2</span><span class="p">,</span> <span class="n">thr2</span> <span class="o">=</span> <span class="n">sklearn_roc_curve</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">predictions</span><span class="p">,</span> <span class="n">sample_weight</span><span class="o">=</span><span class="n">weights</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">fpr1</span><span class="p">,</span> <span class="n">fpr2</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">tpr1</span><span class="p">,</span> <span class="n">tpr2</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">thr1</span><span class="p">,</span> <span class="n">thr2</span><span class="p">)</span>
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<pre>(True, True, True)</pre>
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<h3 id="Weighted-Kolmogorov-Smirnov-distance">Weighted Kolmogorov-Smirnov distance<a class="anchor-link" href="#Weighted-Kolmogorov-Smirnov-distance">&#182;</a></h3><p>ROC curve is an important object by itself and freqently used in ML to compare distribution, 
but the magic is that it contains all order-based information about original distributions.</p>
<p>Next we compute KS distance based on ROC curve:</p>
$$ KS = \max_x \left| F_1(x) - F_2(x) \right| $$<p>(Order-based information == information, which is preserved under monotonic transformations)</p>

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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">ks_2samp</span><span class="p">(</span><span class="n">distribution1</span><span class="p">,</span> <span class="n">distribution2</span><span class="p">,</span> <span class="n">weights1</span><span class="p">,</span> <span class="n">weights2</span><span class="p">):</span>
    <span class="n">labels</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">distribution1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">distribution2</span><span class="p">))</span>
    <span class="n">predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">distribution1</span><span class="p">,</span> <span class="n">distribution2</span><span class="p">])</span>
    <span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">weights1</span><span class="p">,</span> <span class="n">weights2</span><span class="p">])</span>

    <span class="n">fpr</span><span class="p">,</span> <span class="n">tpr</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">roc_curve</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">predictions</span><span class="p">,</span> <span class="n">sample_weight</span><span class="o">=</span><span class="n">weights</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">fpr</span> <span class="o">-</span> <span class="n">tpr</span><span class="p">))</span>
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<div class=" highlight hl-ipython2"><pre><span class="n">data1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
<span class="n">data2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
<span class="n">weights1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span> 
<span class="n">weights2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span> 
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<p><strong>Compare results</strong> with standard function:</p>

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<div class=" highlight hl-ipython2"><pre><span class="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">ks_2samp</span> <span class="k">as</span> <span class="n">ks_2samp_scipy</span>
<span class="k">print</span><span class="p">(</span> <span class="n">ks_2samp</span><span class="p">(</span><span class="n">data1</span><span class="p">,</span> <span class="n">data2</span><span class="p">,</span> <span class="n">weights1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">size</span><span class="p">),</span> <span class="n">weights2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">size</span><span class="p">))</span> <span class="p">)</span>
<span class="k">print</span><span class="p">(</span> <span class="n">ks_2samp_scipy</span><span class="p">(</span><span class="n">data1</span><span class="p">,</span> <span class="n">data2</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="p">)</span>
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<pre>0.416
0.416
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<p>Important moment that written function supports weights, while scipy version doesn't.</p>

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<p><strong>TODO:</strong> optimal F1, optimal accuracy based on ROC curve.</p>

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<h2 id="Conclusion">Conclusion<a class="anchor-link" href="#Conclusion">&#182;</a></h2><p>Many things can be written in numpy in efficient way, moreover resulting code will be short and (surpise!) more readable.</p>
<p>But this skill requires much practice.</p>
<h2 id="See-also">See also<a class="anchor-link" href="#See-also">&#182;</a></h2><ul>
<li>Second part of post: <a href="/2015/09/30/NumpyTipsAndTricks2.html">numpy tips and tricks 2</a></li>
<li>I used example from <a href="http://www.janeriksolem.net/2009/06/histogram-equalization-with-python-and.html">histogram equalization with numpy</a></li>
<li><a href="/2015/09/08/SpeedBenchmarks.html">Numpy log-likelihood benchmark</a></li>
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<p>
    This post was written in IPython. You can download the notebook from
     <a href='https://github.com/arogozhnikov/arogozhnikov.github.io/tree/master/notebooks'>
    repository</a>.
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